The calculation of the Stackelberg–Nash equilibrium as a fixed point problem in static hierarchical games

Abstract

In this paper, a regularized version of the Extraproximal Method together with a Stochastic Approach is suggested to calculate the Stackelberg–Nash equilibrium in a N-person finite game. In this game, two levels of hierarchy in decision making are considered: one leader and ($\hbox {N}-1$) followers. Here, the followers playing according to the Nash equilibrium concept among themselves to the leaders announced strategy. An Extraproximal Technique is used to realize the application of a two-step procedure for finding the Nash equilibrium corresponding for the followers providing the loss function for a leader: at the first (or preliminary) step some predictive approximation of the a current approximation is calculated, at the second step (the main step of the iteration) this prediction is used to complete the current iteration. The next step, applying the stochastic gradient projection technique suggested, the Stackelberg–Nash equilibrium is found. The optimality conditions for a strategy being an equilibrium in participants game are derived based on the strong convexity of the $\delta$-regularized loss function. Simulation results illustrate the feasibility and the performance of this method.